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Partial regularity of minimizers of p ( x )‐growth functionals with 1 < p ( x ) < 2
Author(s) -
Usuba Kunihiro
Publication year - 2015
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/bdv013
Subject(s) - mathematics , combinatorics , hausdorff measure , hausdorff space , measure (data warehouse) , type (biology) , energy (signal processing) , mathematical analysis , hausdorff dimension , computer science , biology , ecology , statistics , database
We prove partial regularity of minimizers u for p ( x ) ‐energy functionals of the following type: E ( u ) = ∫ Ω( A i j α β( x , u ) D α u i D β u j ) p ( x ) / 2d x , assuming thatA i j α β( x , u )and p ( x ) are sufficiently smooth and that p ( x ) is subquadratic. We prove that u ∈ C 0 , α( Ω 0 )for some α ∈ ( 0 , 1 ) and an open setΩ 0 ⊂ Ω withH m - γ 1( Ω - Ω 0 ) = 0 , whereH s denotes the s ‐dimensional Hausdorff measure andγ 1 = inf Ω p ( x ) .